The number of ways of selecting two distinct numbers from the first 15 natural numbers such that their sum is a multiple of 5, is equal to

Options

A. 20
B. 36
C. 21
D. 16

Answer

To find the number of ways of selecting two distinct numbers from the first 15 natural numbers such that their sum is a multiple of 5, we can follow these steps:

### Step 1: Identify the first 15 natural numbers
The first 15 natural numbers are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \]

### Step 2: Determine the possible sums that are multiples of 5
The possible sums that can be formed by two distinct numbers from this set and are multiples of 5 are:
- 5
- 10
- 15
- 20
- 25

### Step 3: Find pairs that sum to each multiple of 5

1. **For a sum of 5:**
- Possible pairs: (1, 4), (2, 3)
- Total distinct pairs: 2

2. **For a sum of 10:**
- Possible pairs: (1, 9), (2, 8), (3, 7), (4, 6)
- Total distinct pairs: 4

3. **For a sum of 15:**
- Possible pairs: (1, 14), (2, 13), (3, 12), (4, 11), (5, 10), (6, 9), (7, 8)
- Total distinct pairs: 7

4. **For a sum of 20:**
- Possible pairs: (5, 15), (6, 14), (7, 13), (8, 12), (9, 11)
- Total distinct pairs: 5

5. **For a sum of 25:**
- Possible pairs: (10, 15)
- Total distinct pairs: 1

### Step 4: Sum the total number of distinct pairs
Now, we add the number of distinct pairs for each case:
\[ 2 \text{ (for 5)} + 4 \text{ (for 10)} + 7 \text{ (for 15)} + 5 \text{ (for 20)} + 1 \text{ (for 25)} = 19 \]

### Conclusion
The total number of ways to select two distinct numbers from the first 15 natural numbers such that their sum is a multiple of 5 is **19**.